Relativistic doppler effect - inconsistency in my derivations

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Hello everyone,

I'm trying to calculate the doppler shift in frequency of a moving source. I'm approaching the problem from two different frames of reference and getting inconsistent results. what am I missing here?

consider the special case of transverse doppler effect (θ=π/2). light reaches observer from y-direction.

reference frame 1: source travelling towards x-direction. observer at rest.
Δto=1/fo - observer's time between subsequent wave crests.
due to time dilation, source's time should be running slower Δ[itex]\tau[/itex]s=Δto/[itex]\gamma[/itex], so actual frequency of source is 1/Δ[itex]\tau[/itex]s=fs=fo*[itex]\gamma[/itex] (the correct relation)

reference frame 2: observer travelling towards x-direction. source at rest.
Δts=1/fs - source's time between subsequent wave crests.
due to time dilation, observer's time should be running slower Δ[itex]\tau[/itex]o=Δts/[itex]\gamma[/itex], so frequency observed is 1/Δ[itex]\tau[/itex]o=fo=[itex]\gamma[/itex]fs
which leaves actual frequenct of source fs=fo/[itex]\gamma[/itex]

why are these two results seemingly inconsistent?

thanks in advance.
 

Answers and Replies

  • #2
ghwellsjr
Science Advisor
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You are doing the same thing in both frames but simply renaming the source and the observer which is the correct thing to do and is showing the reciprocal relationship between the source and the observer but you should have left the second equation as fo=fs*γ. The actual frequency of both the source and the observer is the same in their own rest frame and they each see the same Doppler frequency shift of the other one. Isn't that what your equations indicate? And isn't that the whole point of Relativistic Doppler?
 
  • #3
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Hello everyone,

I'm trying to calculate the doppler shift in frequency of a moving source. I'm approaching the problem from two different frames of reference and getting inconsistent results. what am I missing here?

consider the special case of transverse doppler effect (θ=π/2). light reaches observer from y-direction.

reference frame 1: source travelling towards x-direction. observer at rest.
Δto=1/fo - observer's time between subsequent wave crests.
due to time dilation, source's time should be running slower Δ[itex]\tau[/itex]s=Δto/[itex]\gamma[/itex], so actual frequency of source is 1/Δ[itex]\tau[/itex]s=fs=fo*[itex]\gamma[/itex] (the correct relation)

reference frame 2: observer travelling towards x-direction. source at rest.
Δts=1/fs - source's time between subsequent wave crests.
due to time dilation, observer's time should be running slower Δ[itex]\tau[/itex]o=Δts/[itex]\gamma[/itex], so frequency observed is 1/Δ[itex]\tau[/itex]o=fo=[itex]\gamma[/itex]fs
which leaves actual frequenct of source fs=fo/[itex]\gamma[/itex]

why are these two results seemingly inconsistent?

thanks in advance.
Here is a paper which might be helpful to you:

The relativistic Doppler effect: when a zero frequency shift or a red shift exists for sources approaching the observer
Annalen der Physik (Berlin) 523, No. 3, 239 - 246 (2011); http://arxiv.org/abs/1006.4407
 
  • #4
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Do you know, a light source, when it is approaching (moving closer to) the observer, may cause a red shift?
 
  • #5
3,872
88
Hello everyone,

I'm trying to calculate the doppler shift in frequency of a moving source. I'm approaching the problem from two different frames of reference and getting inconsistent results. what am I missing here?

consider the special case of transverse doppler effect (θ=π/2). light reaches observer from y-direction. [..]
Hi, what you missed is rather basic: the angle of light propagation wrt frame 2 is not equal to that wrt frame 1. Thus you must:
1. specify in which frame θ = 90 degrees
2. calculate the angle in the other frame, and account for the corresponding Doppler effect.
 

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